Regular Hjelmslev Planes . Ii

In this paper we introduce the notion of an «-partition for a Hjelmslev-matrix (//-matrix). This allows us to prove a new composition theorem for //-matrices. We obtain the existence of (/, r)//-matrices and hence of regular (/, ry-H-phaes for infinitely many series of invariants which were not yet known. In fact, many of these invariants were not even known to occur as the invariants of any //-plane at all (whether regular or not). 0. Introduction. In 1976, the author introduced the notion of a 'regular' projective Hjelmslev plane (briefly: regular //"-plane) [4]; such //"-planes admit a nice abelian collineation group. We proved that there are regular //"-planes with invariants (qn, q) for every prime power q. We could also show that there are regular (/, r)-//"-planes (where / is not a power of r) for all the invariant pairs of //"-planes that had been constructed by Drake and Lenz in [3]. Finally, we could also obtain some invariant pairs (/, r), that previously were not even known to be the invariants of an //"-plane at all (whether regular or not). To prove these results, the notions of //"-matrices and x-choices for Hmatrices were essential, allowing certain composition methods. In this paper we want to introduce the notion of an x-partition for an //"-matrix; we will then prove that certain (q", ^-//"-matrices possess (qn~l + l)-partitions. We prove a composition theorem using this notion; this will enable us to obtain regular (/, r)-H-planes for many new invariant pairs. In particular, we will be able to obtain many of the invariant pairs (/, r) constructed in [2]; but we will also obtain infinitely many pairs totally unknown up to now. These results will be improved in the last section by constructing (q", q)-//"-matrices with even better partitions. 1. Preliminary knowledge. We refer the reader to the literature for the definition of projective Hjelmslev planes (//"-planes) and for the invariants of Received by the editors December 1, 1976 and, in revised form, March 21, 1977. AMS (MOS) subject classifications (1970). Primary 05B25, 05B10; Secondary 50A99, 50D99.